The double layer model is used to visualize the ionic
environment in the vicinity of a charged surface. It can be
either a metal under potential or due to ionic groups on the
surface of a dielectric. It is easier to understand this model as
a sequence of steps that would take place near the surface if its
neutralizing ions were suddenly stripped away.

One of the first principles which we must be recognized is
that matter at the boundary of two phases possesses properties
which differentiates it from matter freely extended in either of
the continuous phases separated by the interface. When talking
about a solid-solution interface, it is perhaps easier to
visualize a difference between the interface and the solid than
it is to visualize a difference between the interface and the
extended liquid phase. Where we have a charged surface, however,
there must be a balancing counter charge, and this counter charge
will occur in the liquid. The charges will not be uniformly
distributed throughout the liquid phase, but will be concentrated
near the charged surface. Thus, we have a small but finite
volume of the liquid phase which is different from the extended
liquid. This concept is central to electrochemistry, and
reactions within this interfacial boundary that govern external
observations of electrochemical reactions. It is also of great
importance to soil chemistry, where colloidal particles with
different surface charges play a crucial role.

There are several theoretical treatments of the solid-liquid
interface. We will look at a few common ones, not so much from
the position of needing to use them, but more from the point of
what they can tell us about the soil surface.

A. Helmholtz Double Layer

This theory is a simplest approximation that the surface
charge is neutralized by opposite sign counterions placed at an
increment of d away from the surface.

The surface charge potential is linearly dissipated from the
surface to the contertions satisfying the charge. The distance, d,
will be that to the center of the countertions, i.e. their
radius. The Helmholtz theoretical treatment does not adequately
explain all the features, since it hypothesizes rigid layers of
opposite charges. This does not occur in nature.

B. Gouy-Chapman Double Layer

Gouy suggested that interfacial potential at the charged
surface could be attributed to the presence of a number of ions
of given sign attached to its surface, and to an equal number of
ions of opposite charge in the solution. In other words,
counter ions are not rigidly held, but tend to diffuse into the
liquid phase until the counter potential set up by their
departure restricts this tendency. The kinetic energy of the
counter ions will, in part, affect the thickness of the resulting
diffuse double layer. Gouy and, independently, Chapman
developed theories of this so called diffuse double layer
in which the change in concentration of the counter ions near a
charged surface follows the Boltzman distribution

Already, however, we are in error, since derivation of this
form of the Boltzman distribution assumes that activity is equal
to molar concentration. This may be an OK approximation for the
bulk solution, but will not be true near a charged surface.
Now, since we have a diffuse double layer, rather than a rigid
double layer, we muse concern ourselves with the volume charge
density rather than surface charge density when studying the
coulombic interactions between charges. The volume charge
density, r , of any volume, i, can be
expressed as

ri = Szieni

The coulombic interaction between charges can, then, be
expressed by the Poisson equation. For plane surfaces, this can
be expressed as

d2Y/dx2
= -4pr/d

where Y varies from Yo at the surface to 0 in bulk
solution. Thus, we can relate the charge density at any given
point to the potential gradient away from the surface.
Combining the Boltzmann distribution with the Poisson equation
and integrating under appropriate limits, yields the electric
potential as a function of distance from the surface. The
thickness of the diffuse double layer:

ldouble = [erkT/(4pe2Sniozi2)]1/2

at room temperature can be simplified as

ldouble = 3.3*106er/(zc1/2)

in other words, the double layer thickness decreases with
increasing valence and concentration.

The Gouy-Chapman theory describes a rigid charged surface,
with a cloud of oppositely charged ions in the solution, the
concentration of the oppositely charged ions decreasing with
distance from the surface. This is the so-called diffuse double
layer.

This theory is still not entirely accurate. Experimentally,
the double layer thickness is generally found to be somewhat
greater than calculated. This may relate to the error
incorporated in assuming activity equals molar concentration when
using the desired form of the Boltzman distribution.
Conceptually, it tends to be a function of the fact that both
anions and cations exist in the solution, and with increasing
distance away from the surface the probability that ions of the
same sign as the surface charge will be found within the double
layer increase as well.

C. Stern Modification of the Diffuse double Layer

The Gouy-Chapman theory provides a better approximation of
reality than does the Helmholtz theory, but it still has limited
quantitative application. It assumes that ions behave as point
charges, which they cannot, and it assumes that there is no
physical limits for the ions in their approach to the surface,
which is not true. Stern, therefore, modified the Gouy-Chapman
diffuse double layer. His theory states that ions do have
finite size, so cannot approach the surface closer than a few
nm. The first ions of the Gouy-Chapman Diffuse Double Layer are
not at the surface, but at some distance d
away from the surface. This distance will usually be taken as
the radius of the ion. As a result, the potential and
concentration of the diffuse part of the layer is low enough to
justify treating the ions as point charges.

Stern also assumed that it is possible that some of the ions
are specifically adsorbed by the surface in the plane d, and this layer has become known as the
Stern Layer. Therefore, the potential will drop by Yo - Yd
over the "molecular condenser" (ie. the Helmholtz
Plane) and by Yd over the
diffuse layer. Yd has
become known as the zeta (z)
potential.

This diagram serves as a visual comparison of the amount of
counterions in each the Stearn Layer and the Diffuse Layer of
smectite when saturated with the three alkali earth ions, as in
the above table.

The double layer is formed in order to neutralize the charged
surface and, in turn, causes an electrokinetic potential between
the surface and any point in the mass of the suspending liquid.
This voltage difference is on the order of millivolts and is
referred to as the surface potential. The magnitude of the
surface potential is related to the surface charge and the
thickness of the double layer. As we leave the surface, the
potential drops off roughly linearly in the Stern layer and then
exponentially through the diffuse layer, approaching zero at the
imaginary boundary of the double layer. The potential curve is
useful because it indicates the strength of the electrical force
between particles and the distance at which this force comes into
play. A charged particle will move with a fixed velocity in a
voltage field. This phenomenon is called electrophoresis. The
particle’s mobility is related to the dielectric constant and
viscosity of the suspending liquid and to the electrical
potential at the boundary between the moving particle and the
liquid. This boundary is called the slip plane and is usually
defined as the point where the Stern layer and the diffuse layer
meet. The relationship between zeta potential and surface
potential depends on the level of ions in the solution.The figure
above represents the change in charge density through the diffuse
layer. One shows considered to be rigidly attached to the
colloid, while the diffuse layer is not. As a result, the
electrical potential at this junction is related to the mobility
of the particle and is called the zeta potential. Although zeta
potential is an intermediate value, it is sometimes considered to
be more significant than surface potential as far as
electrostatic repusion is concerned.